This is a big problem. Insulation companies will not remove all your stuff to install new insulation, so effectively by having a load of storage in the loft, stops you from having a grant for additional insulation.
So your choices. Well you are limited a bit, but you can still install insulation in the rafters. You need to leave a 50mm minimum air gap below the felt and down to the eaves to allow for air circulation to reduce the risk of condensation on the underside of the felt. So the easiest way is to attach insulation boards to the underside of the rafters. Most people here would use a phenolic board like Kingspan or Celotex. Personally I would recommend a wood fibre board. Have a look at the post on decrement vs insulation for the reasoning behind this.
This then turns your loft space into a bit of a half way house. Part insulated from below (normally with the existing 100mm of mineral wool or glass fibre) and then insulated from above by the, hopefully, wood fibre boards. This creates a better storage area as it will not suffer as much from the extremes of hot and cold.
Please do not fit the thin reflective insulation unless you are prepared for a lot of careful detailing work, otherwise you are wasting your money. These require conventional insulation behind them (50mm) and two airtight layers above and below the actual reflective insulation. Getting airtightness can be difficult at the best of times let alone twice on the same job.
If you were happy to move stuff around and create clear areas to work in, you could ‘lift’ the floor up, insulate below and then refit the floor. This does add weight to the floor load, so you need to be a bit careful depending on what you are storing, condition of joists, depth of joists etc. It will also make the storage area in the loft prone to greater temperature swings. One could also replace the boards with insulated boards. These are available as chip board with phenolic insulation stuck to the bottom. So same issues as ‘lifting’ the floor apply here.
Don’t think that by having loads of stuff in the loft it actually helps with the insulation, because it doesn’t (unless you are storing lots of duvets up there!) So have a think about what you are prepared to do and then make an informed choice. Remember that modern day standards suggest that you have 270mm (a foot) of fluffy insulation in your ceiling.